System and method for a media codec employing a reversible transform obtained via matrix lifting

ABSTRACT

A system and method for encoding and/or decoding a signal, such as an audio signal, employing a reversible transform obtained via matrix lifting. This reversible transform not only converts integer input to integer output, but also reconstructs the exact input from the output. It is one of the key modules for lossless and progressive to lossless audio codecs. The system and method of the invention produces smaller quantization noise and better compression performance of lossless and progressive to lossless codecs previously known. A number of embodiments employing RMDCT solutions are described. Matrix lifting is used to implement a reversible fast Fourier transform (FFT) and a reversible fractional-shifted FFT, respectively, which are further combined with reversible rotations to form a RMDCT. A progressive-to-lossless embedded audio codec (PLEAC) employing RMDCT is implemented with superior results for both lossless and lossy audio compression.

BACKGROUND

This application claims priority under 35 U.S.C. Section 119(e)(1) ofprovisional application No. 60/513,006 filed Oct. 20, 2003 and entitled“Reversible FFT, Fractional-shifted FFT and MDCT Implementation ViaMatrix Lifting”.

TECHNICAL FIELD

This invention is directed toward a system and method for encoding anddecoding data. More specifically, the invention is directed toward asystem and method for encoding and/or decoding data, such as, forexample audio or video data, by employing a reversible transformobtained via matrix lifting.

BACKGROUND ART

High performance audio codec brings digital music into reality. Popularaudio compression technologies, such as MPEG-1 layer 3 (MP3), MPEG4audio, Real Audio and Windows Media Audio (WMA), are lossy in nature. Inthese compression technologies, the audio waveform is distorted inexchange for a higher compression ratio. In quality criticalapplications such as a professional recording/editing studio, it isimperative to preserve the original audio. That is, the audio should becompressed in a lossless fashion. An especially attractive feature of alossless audio codec is the progressive-to-lossless codec, where theaudio is compressed into a lossless bitstream, which may be furthertruncated at an arbitrary point to provide a lossy bitstream of lesserbitrate without re-encoding. Thus, progressive-to-lossless media codecoffers the greatest flexibility in compression. During initial encoding,the media may be compressed to lossless, which preserves all of theinformation of the original media. Later, if the transmission bandwidthor the storage space is insufficient to accommodate the full losslessmedia, the compressed media bitstream may be effortlessly truncated towhatever bitrate is desired. The state-of-the-art image compressionalgorithm, the JPEG 2000[1], has the progressive-to-lossless compressionmode. However, no existing audio codec operates in theprogressive-to-lossless mode.

A primary reason for the lack of progressive-to-lossless audio codec isdue to the lack of high quality reversible transform. Most losslessaudio coding approaches, such as [8][9][10], are built upon a lossyaudio coder. The audio is first encoded with an existing lossy codec,then the residue error between the original audio and the lossy codedaudio is encoded. The resultant compressed bitstream has two ratepoints, the lossy base bitrate and the lossless bitrate. It may not bescaled at other bitrate points. Since the quantization noise in thelossy coder is difficult to model, such approaches usually lead to adrop in the lossless compression efficiency. Moreover, this codingapproach is also more complex, as it requires the implementation of abase coder and a residue coder. Some other approaches, e.g., [11], buildthe lossless audio coder directly through a predictive filter and thenencode the prediction residue. The approaches may achieve good losslesscompression performance. However, there is still no scalability of theresultant bitstream.

There are many existing schemes for encoding audio files. Several suchschemes attempt to achieve higher compression ratios by using knownhuman psychoacoustic characteristics to mask the audio file. Apsychoacoustic coder is an audio encoder which has been designed to takeadvantage of human auditory masking by dividing the audio spectrum ofone or more audio channels into narrow frequency bands of differentsizes optimized with respect to the frequency selectivity of humanhearing. This makes it possible to sharply filter coding noise so thatit is forced to stay very close in frequency to the frequency componentsof the audio signal being coded. By reducing the level of coding noisewherever there are no audio signals to mask it, and increasing the levelof coding noise wherever there are strong audio signals, the soundquality of the original signal can be subjectively preserved. Usinghuman psychoacoustic hearing characteristics in audio file compressionallows for fewer bits to be used to encode the audio components that areless audible to the human ear. Conversely, more bits can then be used toencode any psychoacoustic components of the audio file to which thehuman ear is more sensitive. Such psychoacoustic coding makes itpossible to greatly improve the quality of an encoded audio at given bitrate.

Psychoacoustic characteristics are typically incorporated into an audiocoding scheme in the following way. First, the encoder explicitlycomputes auditory masking thresholds of a group of audio coefficients,usually a “critical band,” to generate an “audio mask.” These thresholdsare then transmitted to the decoder in certain forms, such as, forexample, the quantization step size of the coefficients. Next, theencoder quantizes the audio coefficients according to the auditory mask.For auditory sensitive coefficients, those to which the human ear ismore sensitive, a smaller quantization step size is typically used. Forauditory insensitive coefficients, those to which the human ear is lesssensitive, a larger quantization step size is typically used. Thequantized audio coefficients are then typically entropy encoded, eitherthrough a Huffman coder such as the MPEG4 AAC quantization and coding, avector quantizer such as the MPEG-4 TwinVQ, or a scalable bitplane codersuch as the MPEG-4 BSAC coder.

In each of the aforementioned conventional audio coding schemes, theauditory masking is applied before the process of entropy coding.Consequently, the masking threshold is transmitted to the decoder asoverhead information. As a result, the quality of the encoded audio at agiven bit rate is reduced to the extent of the bits required to encodethe auditory masking threshold information. Additionally, these audiocoding schemes typically use floating point values in theircalculations. Floating point arithmetic varies across platforms and thuscoding schemes that use floating points are not readily transportableacross these different types of platforms.

Therefore, what is needed is a system and method for encoding ordecoding media data, such as, for example, audio or video data, whereinthe bitstream can be scaled to whatever bitrate is desired. This systemand method should be computationally efficient, while minimizingquantization noise. This encoding and decoding scheme should be portableacross different types of platforms and operate in lossy andprogressive-to-lossless modes.

It is noted that in the remainder of this specification, the descriptionrefers to various individual publications identified by a numericdesignator contained within a pair of brackets. For example, such areference may be identified by reciting, “reference [1]” or simply“[1]”. A listing of the publications corresponding to each designatorcan be found at the end of the Detailed Description section.

SUMMARY

The invention is directed toward a system and method for a codec thatencodes and/or decodes media, such as an audio or video signal,employing a low noise reversible transform. With matrix lifting andmultiple factorization reversible rotation, the quantization noise ofthe reversible transform can be greatly reduced compared to othermethods of encoding and decoding media data. The reversible transformcan be implemented using integer arithmetic, and thus be ported acrossplatforms, as well as scaled from lossless to any desired bitrate.

Embodiments of the system and method according to the invention usematrix lifting to implement a reversible modified discrete cosinetransform (RMDCT) for audio coding. In some embodiments of theinvention, this is done using a reversible Fast Fourier Transform (FFT)or a reversible fractional-shifted FFT, which are further combined withreversible rotations to form the RMDCT.

In one embodiment of the invention, an encoding system comprises areversible transform component obtained via matrix lifting and anentropy encoder. The reversible transform component receives an inputsignal and provides an output of quantized coefficients corresponding tothe input signal. The reversible transform component can employ, forexample, a modified discrete cosine transform (MDCT), a Fast FourierTransform (FFT) or a fractional-shifted FFT to obtain a RMDCT. Thisencoding system can encode data in both lossless andprogressive-to-lossless modes.

Similarly, in another embodiment of the invention, a decoding systemcomprises an entropy decoder and an inverse reversible transformcomponent. The entropy decoder entropy decodes the input bit stream andprovides the decoded information to the inverse transform component. Theinverse transform component then transforms the values from the entropydecoder and provides output values. The inverse transform componentutilizes an inverse transform to essentially revert the computations inthe reversible transform component of the encoder which were obtainedvia matrix lifting.

In yet another embodiment, a progressive-to-lossless embedded audiocodec (PLEAC) employing a RMDCT obtained via matrix lifting isimplemented with superior results for both lossless and lossy audiocompression.

In addition to the just described benefits, other advantages of thepresent invention will become apparent from the detailed descriptionwhich follows hereinafter when taken in conjunction with the drawingfigures which accompany it.

DESCRIPTION OF THE DRAWINGS

The specific features, aspects, and advantages of the invention willbecome better understood with regard to the following description,appended claims, and accompanying drawings where:

FIG. 1 is a diagram depicting a general purpose computing deviceconstituting an exemplary system for implementing the invention.

FIG. 2 is a simplified block diagram of an encoder according to thesystem and method according to the invention.

FIG. 3 is a simplified block diagram of a decoder according to thesystem and method according to the invention.

FIG. 4A depicts a FMDCT via a type-IV DST; while FIG. 4B depicts a FMDCTvia a type-IV DCT.

FIG. 5 depicts a graph of quantization noise versus the rotation angleof different factorization forms: the correspondence between the legendand the factorization forms are: o—(8) x—(31), +—(32) and ⋄—(33). Thebottom solid line is the quantization noise with the combinedfactorization.

FIG. 6 depicts a forward reversible transform obtained via matrixlifting according to the present invention.

FIG. 7 depicts an inverse reversible transform obtained via matrixlifting according to the present invention.

FIG. 8 depicts a flow diagram of a process of using fixed-point float toimplement matrix lifting according to the present invention.

FIG. 9 depicts a flow diagram of a traditional implementation of anormalized FFT wherein a scaling operation is performed after the groupof butterfly operations.

FIG. 10 depicts a flow diagram of a traditional implementation of anormalized FFT wherein a scaling operation is performed before the groupof butterfly operations.

FIG. 11 depicts a flow diagram of an implementation of a normalized FFTaccording to the present invention.

FIG. 12 depicts a progressive-to-lossless audio codec (PLEAC) encoderframework according to the present invention.

FIG. 13 depicts a PLEAC decoder framework according to the presentinvention.

FIG. 14 depicts an exemplary method of encoding a lossless codecaccording to the present invention.

FIG. 15 depicts an exemplary method of decoding the codec that isdepicted in FIG. 10 according to the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the following description of the preferred embodiments of the presentinvention, reference is made to the accompanying drawings that form apart hereof, and in which is shown by way of illustration specificembodiments in which the invention may be practiced. It is understoodthat other embodiments may be utilized and structural changes may bemade without departing from the scope of the present invention.

1.0 Exemplary Operating Environment

FIG. 1 illustrates an example of a suitable computing system environment100 on which the invention may be implemented. The computing systemenvironment 100 is only one example of a suitable computing environmentand is not intended to suggest any limitation as to the scope of use orfunctionality of the invention. Neither should the computing environment100 be interpreted as having any dependency or requirement relating toany one or combination of components illustrated in the exemplaryoperating environment 100.

The invention is operational with numerous other general purpose orspecial purpose computing system environments or configurations.Examples of well known computing systems, environments, and/orconfigurations that may be suitable for use with the invention include,but are not limited to, personal computers, server computers, hand-heldor laptop devices, multiprocessor systems, microprocessor-based systems,set top boxes, programmable consumer electronics, network PCs,minicomputers, mainframe computers, distributed computing environmentsthat include any of the above systems or devices, and the like.

The invention may be described in the general context ofcomputer-executable instructions, such as program modules, beingexecuted by a computer. Generally, program modules include routines,programs, objects, components, data structures, etc. that performparticular tasks or implement particular abstract data types. Theinvention may also be practiced in distributed computing environmentswhere tasks are performed by remote processing devices that are linkedthrough a communications network. In a distributed computingenvironment, program modules may be located in both local and remotecomputer storage media including memory storage devices.

With reference to FIG. 1, an exemplary system for implementing theinvention includes a general purpose computing device in the form of acomputer 110. Components of computer 110 may include, but are notlimited to, a processing unit 120, a system memory 130, and a system bus121 that couples various system components including the system memoryto the processing unit 120. The system bus 121 may be any of severaltypes of bus structures including a memory bus or memory controller, aperipheral bus, and a local bus using any of a variety of busarchitectures. By way of example, and not limitation, such architecturesinclude Industry Standard Architecture (ISA) bus, Micro ChannelArchitecture (MCA) bus, Enhanced ISA (EISA) bus, Video ElectronicsStandards Association (VESA) local bus, and Peripheral ComponentInterconnect (PCI) bus also known as Mezzanine bus.

Computer 110 typically includes a variety of computer readable media.Computer readable media can be any available media that can be accessedby computer 110 and includes both volatile and nonvolatile media,removable and non-removable media. By way of example, and notlimitation, computer readable media may comprise computer storage mediaand communication media. Computer storage media includes both volatileand nonvolatile, removable and non-removable media implemented in anymethod or technology for storage of information such as computerreadable instructions, data structures, program modules or other data.Computer storage media includes, but is not limited to, RAM, ROM,EEPROM, flash memory or other memory technology, CD-ROM, digitalversatile disks (DVD) or other optical disk storage, magnetic cassettes,magnetic tape, magnetic disk storage or other magnetic storage devices,or any other medium which can be used to store the desired informationand which can be accessed by computer 110. Communication media typicallyembodies computer readable instructions, data structures, programmodules or other data in a modulated data signal such as a carrier waveor other transport mechanism and includes any information deliverymedia. The term “modulated data signal” means a signal that has one ormore of its characteristics set or changed in such a manner as to encodeinformation in the signal. By way of example, and not limitation,communication media includes wired media such as a wired network ordirect-wired connection, and wireless media such as acoustic, RF,infrared and other wireless media. Combinations of the any of the aboveshould also be included within the scope of computer readable media.

The system memory 130 includes computer storage media in the form ofvolatile and/or nonvolatile memory such as read only memory (ROM) 131and random access memory (RAM) 132. A basic input/output system 133(BIOS), containing the basic routines that help to transfer informationbetween elements within computer 110, such as during start-up, istypically stored in ROM 131. RAM 132 typically contains data and/orprogram modules that are immediately accessible to and/or presentlybeing operated on by processing unit 120. By way of example, and notlimitation, FIG. 1 illustrates operating system 134, applicationprograms 135, other program modules 136, and program data 137.

The computer 110 may also include other removable/non-removable,volatile/nonvolatile computer storage media. By way of example only,FIG. 1 illustrates a hard disk drive 141 that reads from or writes tonon-removable, nonvolatile magnetic media, a magnetic disk drive 151that reads from or writes to a removable, nonvolatile magnetic disk 152,and an optical disk drive 155 that reads from or writes to a removable,nonvolatile optical disk 156 such as a CD ROM or other optical media.Other removable/non-removable, volatile/nonvolatile computer storagemedia that can be used in the exemplary operating environment include,but are not limited to, magnetic tape cassettes, flash memory cards,digital versatile disks, digital video tape, solid state RAM, solidstate ROM, and the like. The hard disk drive 141 is typically connectedto the system bus 121 through anon-removable memory interface such asinterface 140, and magnetic disk drive 151 and optical disk drive 155are typically connected to the system bus 121 by a removable memoryinterface, such as interface 150.

The drives and their associated computer storage media discussed aboveand illustrated in FIG. 1, provide storage of computer readableinstructions, data structures, program modules and other data for thecomputer 110. In FIG. 1, for example, hard disk drive 141 is illustratedas storing operating system 144, application programs 145, other programmodules 146, and program data 147. Note that these components can eitherbe the same as or different from operating system 134, applicationprograms 135, other program modules 136, and program data 137. Operatingsystem 144, application programs 145, other program modules 146, andprogram data 147 are given different numbers here to illustrate that, ata minimum, they are different copies. A user may enter commands andinformation into the computer 110 through input devices such as akeyboard 162 and pointing device 161, commonly referred to as a mouse,trackball or touch pad. Other input devices (not shown) may include amicrophone, joystick, game pad, satellite dish, scanner, or the like.These and other input devices are often connected to the processing unit120 through a user input interface 160 that is coupled to the system bus121, but may be connected by other interface and bus structures, such asa parallel port, game port or a universal serial bus (USB). A monitor191 or other type of display device is also connected to the system bus121 via an interface, such as a video interface 190. In addition to themonitor, computers may also include other peripheral output devices suchas speakers 197 and printer 196, which may be connected through anoutput peripheral interface 195. Of particular significance to thepresent invention, a camera 163 (such as a digital/electronic still orvideo camera, or film/photographic scanner) capable of capturing asequence of images 164 can also be included as an input device to thepersonal computer 110. Further, while just one camera is depicted,multiple cameras could be included as an input device to the personalcomputer 110. The images 164 from the one or more cameras are input intothe computer 110 via an appropriate camera interface 165. This interface165 is connected to the system bus 121, thereby allowing the images tobe routed to and stored in the RAM 132, or one of the other data storagedevices associated with the computer 110. However, it is noted thatimage data can be input into the computer 110 from any of theaforementioned computer-readable media as well, without requiring theuse of the camera 163.

The computer 110 may operate in a networked environment using logicalconnections to one or more remote computers, such as a remote computer180. The remote computer 180 may be a personal computer, a server, arouter, a network PC, a peer device or other common network node, andtypically includes many or all of the elements described above relativeto the computer 110, although only a memory storage device 181 has beenillustrated in FIG. 1. The logical connections depicted in FIG. 1include a local area network (LAN) 171 and a wide area network (WAN)173, but may also include other networks. Such networking environmentsare commonplace in offices, enterprise-wide computer networks, intranetsand the Internet.

When used in a LAN networking environment, the computer 110 is connectedto the LAN 171 through a network interface or adapter 170. When used ina WAN networking environment, the computer 110 typically includes amodem 172 or other means for establishing communications over the WAN173, such as the Internet. The modem 172, which may be internal orexternal, may be connected to the system bus 121 via the user inputinterface 160, or other appropriate mechanism. In a networkedenvironment, program modules depicted relative to the computer 110, orportions thereof, may be stored in the remote memory storage device. Byway of example, and not limitation, FIG. 1 illustrates remoteapplication programs 185 as residing on memory device 181. It will beappreciated that the network connections shown are exemplary and othermeans of establishing a communications link between the computers may beused.

The exemplary operating environment having now been discussed, theremaining parts of this description section will be devoted to adescription of the program modules embodying the invention.

2.0 A System and Method for a Media Codec Employing a ReversibleTransform Obtained Via Matrix Lifting

The system and method according to the invention is described in detailin the following sections. However, in a most general sense, referringto FIG. 2, a data coder system 200 in accordance with an aspect of thepresent invention is illustrated. The encoding system 200 can encodemedia data, such as, for example, image and/or audio data. For instance,the system 200 can be employed in a vast array of audio and/or documentimage applications, including, but not limited to, digital audiosystems, segmented layered image systems, photocopiers, documentscanners, optical character recognition systems, personal digitalassistants, fax machines, digital cameras, digital video cameras and/orvideo games. The encoding system comprises a reversible transformcomponent obtained via matrix lifting 210 and an entropy encoder 220.The reversible transform component 210 receives an input integer signaland provides an output of integer coefficients corresponding to theinput signal. The reversible transform component 210 can employ, forexample, a modified discrete cosine transform (MDCT), a Fast FourierTransform (FFT) or a fractional-shifted FFT to obtain a RMDCT. Theencoding system can operate in lossless or progressive-to-losslessmodes.

As used in this application, the term “computer component” is intendedto refer to a computer-related entity, either hardware, a combination ofhardware and software, software, or software in execution. For example,a computer component may be, but is not limited to being, a processrunning on a processor, a processor, an object, an executable, a threadof execution, a program, and/or a computer. By way of illustration, bothan application running on a server and the server can be a computercomponent. One or more computer components may reside within a processand/or thread of execution and a component may be localized on onecomputer and/or distributed between two or more computers.

Referring to FIG. 3, a simplified data decoder system 300 in accordancewith an aspect of the present invention is illustrated. The decodingsystem 300 comprises an entropy decoder 310 and an inverse transformcomponent 320. The entropy decoder 310 entropy decodes the input bitstream and provides the decoded integer coefficients to the inversetransform component 320. The inverse transform component 320 transformsthe values from the entropy decoder 310 and provides output values. Theinverse transform component utilizes an inverse transform to essentiallyrevert the computations in the reversible transform component of theencoder which were obtained via matrix lifting. The data decoder system300, the entropy decoder 310 and/or the inverse transform component 320can be computer components as that term is defined herein.

The following sections provide further details of the invention and thederivation thereof. Section 2.1 provides an overview of a reversibletransform. The structure of a Float MDCT, a reversible MDCT and a lownoise reversible rotation achieved through multiple factorizations aredescribed in Section 2.2. Then, in Section 2.3, matrix lifting and itsapplication to the reversible transform is described. A reversible FFTand a reversible fractional-shifted FFT are derived through the matrixlifting, and are used to implement a low noise RMDCT codec. For crossplatform reversibility, the RMDCT is implemented with only the integerarithmetic. A number of integer arithmetic implementation issues areexamined in Section 2.4. An exemplary progressive-to-lossless embeddedaudio codec (PLEAC) that incorporates the RMDCT is described in Section2.5. Exemplary methods of coding/decoding media data according to theinvention are discussed in Section 2.6. Experimental results are shownin Section 3.0.

2.1 Reversible Transforms

To develop a progressive to lossless embedded media codec, there are twokey modules: a reversible transform and a lossless embedded entropycoder. The reversible transform is usually derived from the lineartransform of a traditional media codec. By splitting the lineartransform into a number of modules, and implementing each module with areversible transform module, one can construct a reversible transformwhose output resembles that of the linear transform, except for therounding errors. The reversible transform establishes a one-to-onecorrespondence between its input and output, and converts the integerinput to a set of integer coefficients. The lossless embedded entropycoder then encodes the resultant coefficients progressively all the wayto lossless, often in a sub-bitplane by sub-bitplane fashion. Byincorporating both modules in the media codec, one can achieveprogressive-to-lossless coding. If the entire compressed bitstream isdelivered to the decoder, it may exactly reconstruct the original media.If the bitstream is truncated at certain bitrate, the decoder mayreconstruct a high perceptual quality media at that bitrate. The systemand method of the present invention focuses on the design of thereversible transform. A number of conventional embedded entropy coderscan be used with the invention.

For a transform to be reversible, it must convert integer input tointeger output, and be able to exactly reconstruct the input from theoutput. These two properties are essential to guarantee reversibility.Nevertheless, there are other desired properties of the reversibletransform. Low computational complexity is certainly one of them.Another desired property is the normalization. Consider the followingtwo reversible transforms, which are the candidates of the stereo mixerused in the progressive-to-lossless audio codec (PLEAC): $\begin{matrix}\left\{ {\begin{matrix}{{{step}\quad 1\text{:}\quad x^{\prime}} = {x + y}} \\{{{step}\quad 2\text{:}\quad y^{\prime}} = {x - y}}\end{matrix},} \right. & (1) \\{and} & \quad \\\left\{ {\begin{matrix}{{{step}\quad 1\text{:}\quad y^{\prime}} = {x - y}} \\{{{step}\quad 2\text{:}\quad x^{\prime}} = {x - \left\lbrack {0.5y^{\prime}} \right\rbrack}}\end{matrix},} \right. & (2)\end{matrix}$where [·] is a rounding to integer operation, x and y are integerinputs, and x′ and y′ are integer outputs. Both transforms arereversible, however, the output of transform (1) generates a sparseoutput set because all points with x′+y′ equal to odd are not occupied.In comparison, the output of transform (2) is dense.

One notices that if the rounding operation in (2) is removed, it willbecome a linear transform. In general, let a reversible transform be:y=rev(x),  (3)where x is the input integer vector, y is the output integer vector,rev( ) denotes the reversible transform operator. If all of the roundingoperators in the transform are omitted, the transform can be convertedinto a linear transform represented with matrix multiplication:y′=Mx,  (4)where the matrix M is called the characterization matrix of thereversible transform. The characterization matrixes of the reversibletransforms (1) and (2) are: $\begin{matrix}{{M_{0} = \begin{bmatrix}1 & 1 \\1 & {- 1}\end{bmatrix}},{{{and}\quad M_{1}} = \begin{bmatrix}0.5 & 0.5 \\1 & {- 1}\end{bmatrix}},{{respectively}.}} & (5)\end{matrix}$

If x is the set of all possible input data points, the output of thereversible transform occupies a volume roughly determined by det(M)∥X∥,where ∥x∥ is the volume of the input data set, and det(M) is thedeterminant of matrix M. A valid reversible transform cannot have acharacterization matrix with determinant det(M) smaller than 1, becausesuch transform will compact the data, and cause multiple input integervectors mapping to one output integer vector, which contradicts thereversibility. A reversible transform with determinant det(M) greaterthan 1 expands the input data set, and creates holes in the output data.It is extremely difficult to design a lossless entropy coder to dealwith the holes in the output dataset, particularly if the reversibletransform is complex. As a result, a desired property of the reversibletransform is that the determinant of its characterization matrix det(M)is one, i.e., the reversible transform is normalized.

In audio compression, a good linear transform M is already known, whichis the Float MDCT (FMDCT). This can be used to design a RMDCT whosecharacterization matrix is FMDCT. A common strategy of the reversibletransform design is to factor the original linear transform into aseries of simple modules, $\begin{matrix}{{M = {\prod\limits_{i = 1}^{n}M_{i}}},} & (6)\end{matrix}$where one may find a reversible transform for each module M_(i). Forsuch reversible transform design, another desired property concerns thequantization noise of the reversible transform, which is the deviationof the reversible transform from the output of the linear transform ofits characterization matrix:e=rev(x)−Mx,  (7)

The quantization noise e results from the rounding errors of variousstages of the reversible transform. This quantization noise isunavoidable, because it is the byproduct of reversibility, which forcesthe intermediate result and the output to be integers. The roundingerror in each stage of the reversible transform can be considered as anindependent random variable with no correlation with the input andoutput of that stage, thus the aggregated quantization noise of thereversible transform also has likewise little correlation with the inputand the output. Put it in another way, the output of the reversibletransform rev(x) can be considered as the sum of the output of thelinear transform Mx and a random quantization noise e. Apparently, therandom quantization noise increases the entropy of the output. As aresult, less quantization noise leads to better lossless compressionperformance. Because the quantization noise also creates a noise floorin the output of the reversible transform, which reduces the audioquality in the progressive-to-lossless stage, reduction of thequantization noise improves the lossy compression performance as well.The correlation between the quantization noise level of the reversibletransform and its lossless and lossy compression performance isconfirmed by experiments to be discussed later. An ideal reversibletransform therefore should have as low quantization noise as possible.

A FMDCT can be factored into a series of rotations. One way to derive aRMDCT is thus to convert each and every rotation into a reversiblerotation, as shown in [4]. It is common knowledge that a normalizedrotation can be factored into a 3-step lifting operation via:$\begin{matrix}{\left\lbrack \quad\begin{matrix}{\cos\quad\theta} & {{- \sin}\quad\theta} \\{\sin\quad\theta} & {\cos\quad\theta}\end{matrix} \right\rbrack = {{\left\lbrack \quad\begin{matrix}1 & \frac{{\cos\quad\theta} - 1}{\sin\quad\theta} \\0 & 1\end{matrix} \right\rbrack \cdot \left\lbrack \quad\begin{matrix}1 & 0 \\{\sin\quad\theta} & 1\end{matrix} \right\rbrack \cdot \left\lbrack \quad\begin{matrix}1 & \frac{{\cos\quad\theta} - 1}{\sin\quad\theta} \\0 & 1\end{matrix} \right\rbrack}\quad.}} & (8)\end{matrix}$By using roundings in each of the lifting step, the rotation becomesreversible: $\begin{matrix}\left\{ {\begin{matrix}{{{step}\quad 0\text{:}\quad z} = {x + \left\lbrack {c_{0}y} \right\rbrack}} \\{{{step}\quad 1\text{:}\quad x^{\prime}} = {y + \left\lbrack {c_{1}z} \right\rbrack}} \\{{{step}\quad 2\text{:}\quad y^{\prime}} = {z + \left\lbrack {c_{0}x^{\prime}} \right\rbrack}}\end{matrix},} \right. & (9)\end{matrix}$where c₀=(cos θ-l)/sin θ and c_(i)=sin θ are lifting parameters.Existing research on reversible DCT[3] and RMDCT[4] uses thefactorization in (9) as the basic operation for the reversibletransform. Though reversibility is achieved, the quantization noise ofthe approach (8) can be fairly large, and may lead to poor signalrepresentation, and poor lossless and lossy compression performance.

An alternative method to obtaining a RMDCT as discussed in the paragraphabove is to factor a large component of the linear transform M intoupper and lower unit triangular matrixes (UTM), which are triangularmatrixes with diagonal entries 1 or −1. It is shown in [11] that an evensized real matrix M with determinant of norm 1 can be factored into:M=PL _(l) UL ₂,  (10)where P is a permutation matrix, L₁ and L₂, are lower UTMs, and U is anupper UTM. Matrixes L₁, L₂ and U can be reversibly implemented vialifting with N rounding operations, with N being the size of the matrix.The implementation of (10) leads to less rounding operations, and thussmaller quantization noise. Nevertheless, unlike a structured transformsuch as FFT, there is usually no structure in matrix L₁, L₂ and U, andthus there is no fast algorithm to compute the multiplication by matrixL₁, L₂ and U. The computational complexity of the UTM factorizationapproach is hence high.2.2 Float Modified Discrete Cosine Transform (MDCT) and Reversible MDCTI —Reversible Rotation Through Multiple Factorizations

Another method of encoding a media signal to obtain a reversibletransform using reversible rotation thru multiple factorization isdescribed in co-pending patent application Ser. No. 10/300,995 filed onNov. 21, 2002 and entitled “A Progressive to Lossless Embedded AudioCoder (PLEAC) with Multiple Factorization Reversible Transform” by thesame inventor. The idea is to develop a RMDCT from the FMDCT withlow-noise reversible rotations. The float MDCT (FMDCT) can be expressedas:MDCT _(2N) H _(2N),  (11)where $\begin{matrix}{{{MDCT}_{2N} = \left\{ {\sqrt{\frac{2}{N}}\cos\quad\frac{\pi}{N}\left( {i + \frac{1}{2}} \right)\left( {j + \frac{N + 1}{2}} \right)} \right\}_{\begin{matrix}{{i = 0},1,\cdots\quad,{N - 1},} \\{{j = 0},1,\cdots\quad,{{2N} - 1.}}\end{matrix}}},} & (12)\end{matrix}$  and H _(2N) =diag{h(n)}_(n=0, 1, . . . , 2N-1).  (13)MDCT_(2N) is the MDCT matrix, and h(n) is a window function. In MP3audio coding, the window function h(n) is: $\begin{matrix}{{h(n)} = {\sin\quad\frac{\pi}{2N}{\left( {n + 0.5} \right).}}} & (14)\end{matrix}$

According to [12], the FMDCT can be calculated via a type-IV discretesine transform (DST) shown in FIG. 4A. The input signal is first groupedinto pairs of x(n) and x(N-n), x(N+n) and x(2N-n). Each pair is thentreated as a complex number and rotated according to an angle specifiedby h(n). This is called the window rotation. The middle section of thesignal is then transformed by a type-IV DST with: $\begin{matrix}{{{DSTIV}_{N} = \left\lbrack {\sqrt{\frac{2}{N}}{\sin\left( {\frac{\pi}{N}\left( {i + 0.5} \right)\left( {j + 0.5} \right)} \right)}} \right\rbrack_{i,{j = 0},1,\cdots\quad,{N - 1}}},} & (15)\end{matrix}$

The sign of the odd index coefficients are then changed. A 2N-pointtype-IV DST can be further converted to an N-point complexfractional-shifted fast Fourier transform (FFT) with α=β=0.25, as:DSTIV _(2N) =P _(2N) F _(N)(0.25,0.25)Q _(2N) ^(DST) P _(2N),  (16)where: $\begin{matrix}{{P_{2N} = \left\lbrack p_{i,j} \right\rbrack},{{{with}\quad p_{i,j}} = \left\{ {\begin{matrix}1 & {i = {j\quad{and}\quad i\quad{is}\quad{even}}} \\1 & {i = {{2N} - {j\quad{and}\quad i\quad{is}\quad{odd}}}} \\0 & {otherwise}\end{matrix},} \right.}} & (17) \\{Q_{2N}^{DST} = {\begin{bmatrix}\begin{bmatrix}\quad & 1 \\1 & \quad\end{bmatrix} & \quad & \quad \\\quad & ⋰ & \quad \\\quad & \quad & \begin{bmatrix}\quad & 1 \\1 & \quad\end{bmatrix}\end{bmatrix}.}} & (18)\end{matrix}$

The fractional-shifted FFT F_(N)(α,β) is in the form of: $\begin{matrix}{{{F_{N}\left( {\alpha,\beta} \right)} = {\frac{1}{\sqrt{N}}\left\lbrack w_{N}^{{({i + \alpha})}{({j + \beta})}} \right\rbrack}_{i,{j = 0},1,\cdots\quad,{N - 1}}},{{{with}\quad w_{N}} = {\mathbb{e}}^{{- j}\quad 2\quad{\pi/N}}}} & (19)\end{matrix}$where α and β are shifting parameters, and W_(N) is a complex rotation.Note that the fractional-shifted FFT F_(N)(α,β) is a complex matrix,while the other matrixes in equation (16) are real matrixes. This isinterpreted by expanding every element of a complex matrixC=└C_(i, j)┘_(j=0, 1, . . . , N-1) into a 2×2 sub-matrix of the form:$\begin{matrix}{{C = \begin{bmatrix}{re}_{i,j} & {- {im}_{i,j}} \\{im}_{i,j} & {re}_{i,j}\end{bmatrix}_{i,{j = 0},1,\cdots\quad,{N - 1}}},} & (20)\end{matrix}$where re_(i, j) and im_(i, j) are the real and imaginary part of complexvalue c_(i, j), respectively.

Like FFT, the fractional-shifted FFT is an orthogonal transform. Thiscan be easily verified as the Hermitian inner product of any two vectorsof the fractional-shifted FFT is a delta function: $\begin{matrix}\begin{matrix}{{\frac{1}{N}{\sum\limits_{j}{w_{N}^{{- {({i + \alpha})}}{({j + \beta})}} \cdot w_{N}^{{+ {({k + \alpha})}}{({j + \beta})}}}}} = {\frac{1}{N}w_{N}^{{({k - i})}\beta}{\sum\limits_{j}w_{N}^{{({k - i})}j}}}} \\{= {{\delta\left( {k - i} \right)}.}}\end{matrix} & (21)\end{matrix}$

As a corollary, the inverse of the fractional-shifted FFT is:$\begin{matrix}{{{F_{N}^{- 1}\left( {\alpha,\beta} \right)} = {\frac{1}{\sqrt{N}}\left\lbrack w_{N}^{{- {({i + \beta})}}{({j + \alpha})}} \right\rbrack}_{i,{j = 0},1,\cdots\quad,{N - 1}}},} & (22)\end{matrix}$

The fractional-shifted FFT can be decomposed into a pre-rotationΛ_(N)(α,0), FFT F_(N) and a post-rotation Λ_(N)(β,α). Thefractional-shifted FFT can thus be implemented via a standard FFT, as:F _(N) (α,β)=Λ_(N) (β,α)F _(N)Λ_(N)(α, 0),  (23)whereΛ_(N)(α,β)=diag{w _(n) ^(β(j+β))}_(j=0, 1 . . . , N-1),  (24)is a diagonal matrix of N rotations, and $\begin{matrix}{{F_{N} = {\frac{1}{\sqrt{N}}\left\lbrack w_{N}^{ij} \right\rbrack}_{i,{j = 0},1,\cdots\quad,{N - 1}}},.} & (25)\end{matrix}$is the standard FFT.

One notices that the matrix P_(2N) and Q_(2N) ^(DST) are permutationmatrixes and may be implemented as such in the reversible transform. Toderive the RMDCT from the FMDCT above, one simply needs to turn thewindow rotation h(n) into the reversible rotation, and implement thefractional-shifted FFT F_(N)(0.25,0.25) with a reversiblefractional-shifted FFT.

An alternative implementation of FMDCT is to first group signal intopairs of x(n) and x(N+n), rotate them according to an angle specified byh(n), and then transform the middle section of the signal through atype-IV DCT with: $\begin{matrix}{{{DCTIV}_{N} = \left\lbrack {\sqrt{\frac{2}{N}}{\cos\left( {\frac{\pi}{N}\left( {i + 0.5} \right)\left( {j + 0.5} \right)} \right)}} \right\rbrack_{i,{j = 0},1,\cdots\quad,{N - 1}}},} & (26)\end{matrix}$

The implementation can be shown in FIG. 4B. It is easily verified that a2N-point type-IV DCT can be converted to an N-point inversefractional-shifted FFT:DCTIV _(2N) =P _(2N) F _(N) ⁻¹(0.25,0.25)Q _(2N) ^(DCT)P_(2N),  (27)$\begin{matrix}{{{with}\quad Q_{2N}^{DCT}} = {\begin{bmatrix}\begin{bmatrix}1 & \quad \\\quad & {- 1}\end{bmatrix} & \quad & \quad & \quad \\\quad & \begin{bmatrix}1 & \quad \\\quad & {- 1}\end{bmatrix} & \quad & \quad \\\quad & \quad & ⋰ & \quad \\\quad & \quad & \quad & \begin{bmatrix}1 & \quad \\\quad & {- 1}\end{bmatrix}\end{bmatrix}.}} & (28)\end{matrix}$

With FMDCT, the two implementations of FIGS. 4A and 4B lead to the sameresult. However, they lead to slightly different derived reversibletransforms. The FMDCT transform has other alternative forms andimplementations, with different phases and window functions. Somealternative FMDCTs, termed modulated lapped transform (MLT), are shownin [12]. Nevertheless, all FMDCT and alternative forms can be decomposedinto window rotations and the subsequent type-IV DST/DCT. In one casethe RMDCT is derived from the FMDCT in the form of FIG. 4A.Nevertheless, the result can be easily extended to the RMDCT derivedfrom the other FMDCT forms. For example, if an alternative form FMDCTuses the type-IV DCT implementation, one only need to implement theinverse fractional-shifted FFT F_(N) ⁻¹(0.25,0.25), instead of theforward fractional-shifted FFT.

In FIG. 4A, it is shown that the FMDCT consists of the window rotationh(n), the type-IV DST and the sign change. The type-IV DST can beimplemented via a fractional-shifted FFT, which in turn consists of thepre-rotation, the FFT and the post-rotation. The FFT can be implementedvia butterflies; more specifically, the 2N-point FFT can be implementedvia first applying the N-point FFT on the odd and even index signal, andthen combining the output via the butterfly: $\begin{matrix}{{F_{2N} = {{B_{2N}\begin{bmatrix}F_{N} & \quad \\\quad & F_{N}\end{bmatrix}}{OE}_{2N}}},{{{with}\quad B_{2N}} = {\begin{bmatrix}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\\frac{1}{\sqrt{2}} & {- \frac{1}{\sqrt{2}}}\end{bmatrix}\begin{bmatrix}I_{N} & \quad \\\quad & {- {\Lambda_{N}\left( {0.5,0} \right)}}\end{bmatrix}}}} & (29)\end{matrix}$where F_(2N) and F_(N) are the 2N- and N-point FFT, OE_(2N) is apermutation matrix that separates the 2N complex vector into the size-Nvector of even indexes and the size-N vector of odd indexes, and B_(2N)is the butterfly operator. Note in standard FFT([2] Chapter 12), thebutterfly operator is: $\begin{matrix}{{B_{2N}^{\prime} = {\begin{bmatrix}1 & 1 \\1 & {- 1}\end{bmatrix}\begin{bmatrix}I_{N} & \quad \\\quad & {- {\Lambda_{N}\left( {0.5,0} \right)}}\end{bmatrix}}},} & (30)\end{matrix}$and a normalizing operation of 1/{square root}{square root over (N)} isapplied after the entire FFT has been completed. However, this is notfeasible in the reversible FFT, as normalizing by 1/{square root}{squareroot over (N)} is not reversible. One thus needs to adopt (29) as thebasic butterfly. The matrix $\quad\begin{bmatrix}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\\frac{1}{\sqrt{2}} & {- \frac{1}{\sqrt{2}}}\end{bmatrix}$can be considered as the conjugated rotation of −π/4, and Λ_(N)(0.5,0)are N complex rotations, the butterfly operation can thus be implementedas two consecutive rotations. As a result, the entire FMDCT can beimplemented via a series of rotations. By implementing each and everyrotation through the 3-step lifting operation of (8) and (9), one canderive one implementation of the RMDCT. The problem of such animplementation is that the quantization noise of certain rotation anglescould be fairly large, which leads to large quantization noise of theRMDCT.

It is possible to factor the rotation operation with multiple forms. Itis noticed that (8) is not the only form in which a reversible rotationmay be factorized. There are three other factorizations of the rotationin the form of: $\begin{matrix}\begin{matrix}{\begin{bmatrix}{\cos\quad\theta} & {{- \sin}\quad\theta} \\{\sin\quad\theta} & {\cos\quad\theta}\end{bmatrix} = {{\begin{bmatrix}0 & 1 \\{- 1} & 0\end{bmatrix}\begin{bmatrix}1 & \frac{{{- \sin}\quad\theta} - 1}{\cos\quad\theta} \\0 & 1\end{bmatrix}} \cdot \begin{bmatrix}1 & 0 \\{\cos\quad\theta} & 1\end{bmatrix} \cdot}} \\{\begin{bmatrix}1 & \frac{{{- \sin}\quad\theta} - 1}{\cos\quad\theta} \\0 & 1\end{bmatrix}}\end{matrix} & (31) \\\begin{matrix}{\begin{bmatrix}{\cos\quad\theta} & {{- \sin}\quad\theta} \\{\sin\quad\theta} & {\cos\quad\theta}\end{bmatrix} = {{\begin{bmatrix}{- 1} & 0 \\0 & 1\end{bmatrix}\begin{bmatrix}1 & \frac{{\sin\quad\theta} - 1}{\cos\quad\theta} \\0 & 1\end{bmatrix}} \cdot \begin{bmatrix}1 & 0 \\{\cos\quad\theta} & 1\end{bmatrix} \cdot}} \\{\begin{bmatrix}1 & \frac{{\sin\quad\theta} - 1}{\cos\quad\theta} \\0 & 1\end{bmatrix}\begin{bmatrix}0 & 1 \\1 & 0\end{bmatrix}}\end{matrix} & (32) \\\begin{matrix}{\begin{bmatrix}{\cos\quad\theta} & {{- \sin}\quad\theta} \\{\sin\quad\theta} & {\cos\quad\theta}\end{bmatrix} = {{\begin{bmatrix}{- 1} & 0 \\0 & 1\end{bmatrix}\begin{bmatrix}1 & \frac{{{- \cos}\quad\theta} - 1}{\sin\quad\theta} \\0 & 1\end{bmatrix}} \cdot \begin{bmatrix}1 & 0 \\{\sin\quad\theta} & 1\end{bmatrix} \cdot}} \\{\begin{bmatrix}1 & \frac{{{- \cos}\quad\theta} - 1}{\sin\quad\theta} \\0 & 1\end{bmatrix}\begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix}}\end{matrix} & (33)\end{matrix}$

The core of the factorization is still the 3-step lifting operation of(9). However, the pair of the input/output variables may be swappedbefore (as in (32)) and after (as in (31)) the lifting operation. Thesign of the input/output may be changed as well. The additional forms offactorization lead to different lifting parameters c₀ and c₁, for thesame rotation angle θ, with certain form of factorization has a lowerquantization noise than the others.

In the following, the optimal factorization form for different rotationangle θ that achieves the lowest quantization noise in the mean squareerror (MSE) sense is selected. Let Δx′ and Δy′ be the quantization noiseof the reversible rotation. The goal is to minimize the MSE:E[Δx′²]+E[Δy′²]. One notices that it is the rounding operation thatintroduces the quantization noise into the reversible transform. Thecoefficient swapping and sign changing operations do not introduceadditional quantization noise. Let Δ be the quantization noise of onerounding operation:[x]=x+Δ,  (34)

One may model the quantization noise in the reversible transform as:$\begin{matrix}{\begin{bmatrix}{\Delta\quad x^{\prime}} \\{\Delta\quad y^{\prime}}\end{bmatrix} = \begin{bmatrix}{{c_{1}\Delta_{0}} + \Delta_{1}} \\{{\left( {{c_{0}c_{1}} + 1} \right)\Delta_{0}} + {c_{0}\Delta_{1}} + \Delta_{2}}\end{bmatrix}} & (35)\end{matrix}$where Δ₀, Δ₁ and Δ₂ are the quantization noise at the lifting steps 0, 1and 2 of equation (9), respectively. Assuming the quantization noise ateach step being independent and identically distributed, with E[Δ²]being the average energy of the quantization noise of a single roundingoperation, the MSE of the quantization noise of the reversible transformcan be calculated as:E[Δx′ ² ]+E[Δy′ ²]={(1+c₀c₁)²+C₀ ²+c₁ ²+2}E[Δ ²]  (36)

The quantization noise versus the rotation angles for differentfactorization forms (8), (31)-(33) is plotted in FIG. 5. One observesthat with any single factorization, the quantization noise can be fairlylarge at certain rotation angles. By switching among differentfactorization forms, or more specifically, by using factorization forms(8), (31), (32) and (33) for rotation angles (−0.25π,0.25π),(−0.750π,−0.25π), (0.25π,0.75π) and (0.75 π,1.25π), respectively, onemay control the quantization noise to be at most 3.2E[Δ²]. The magnitudeof E[Δ²] depends on the rounding operation used. If one uses roundingtowards the nearest integer, the average energy of the quantizationnoise of one rounding step is: $\begin{matrix}{{{E\left\lbrack {\Delta\quad x^{2}} \right\rbrack} = \frac{1}{12}},} & (37)\end{matrix}$

If rounding towards zero is used, the average energy becomes:$\begin{matrix}{{{E\left\lbrack {\Delta\quad x^{2}} \right\rbrack} = \frac{1}{3}},} & (38)\end{matrix}$

It is apparent that the rounding towards the nearest integer ispreferred, as it generates smaller quantization noise per roundingoperation.

By using the multiple factorization reversible rotation to replace eachrotation in the FMDCT, one may derive a RMDCT with relatively low noisethan simply using the reversible rotation of form (8). However, one canfurther improve upon this scheme by using matrix lifting as discussedbelow.

2.3 Reversible MDCT II—Matrix Lifting

In this section, as an extension to the techniques discussed above, itis shown that it is possible to derive a reversible transform to beemployed with the system and method of the invention through matrixlifting, which will further lower the quantization noise over theresults obtained by using multiple factorization reversible rotationalone.

2.3.1 Matrix Lifting

Theory 1: Every non-singular even sized matrix S_(2N) (real or complex)of size 2N×2N can be factored into: $\begin{matrix}{{S_{2N} = {{{{{P_{2N}\begin{bmatrix}I_{N} & \quad \\A_{N} & I_{N}\end{bmatrix}}\begin{bmatrix}B_{N} & \quad \\\quad & I_{N}\end{bmatrix}}\begin{bmatrix}I_{N} & C_{N} \\\quad & I_{N}\end{bmatrix}}\begin{bmatrix}I_{N} & \quad \\D_{N} & I_{N}\end{bmatrix}}Q_{2N}}},} & (39)\end{matrix}$where P_(2N) and Q_(2N) are permutation matrixes of size 2N×2N, I_(N) isthe identity matrix, A_(N), C_(N) and D_(N) are N×N matrixes, and B_(N)is a non-singular N×N matrix.

Proof: Since S_(2N) is non-singular, there exist permutation matrixesP_(2N) ^(t) and Q_(2N) ^(t) so that $\begin{matrix}{{S_{2N} = {{P_{2N}\begin{bmatrix}S_{11} & S_{12} \\S_{21} & S_{22}\end{bmatrix}}Q_{2N}}},} & (40)\end{matrix}$with S₁₂ being non-singular. Observing that: $\begin{matrix}{{{\begin{bmatrix}S_{11} & S_{12} \\S_{21} & S_{22}\end{bmatrix}\begin{bmatrix}I_{N} & \quad \\{{- S_{12}^{- 1}}S_{11}} & I_{N}\end{bmatrix}} = \begin{bmatrix}\quad & S_{12} \\{- \left( {{S_{22}S_{12}^{- 1}S_{11}} - S_{21}} \right)} & S_{22}\end{bmatrix}},} & (41)\end{matrix}$by taking determinant of S_(2N), and using the distributive property ofthe determinant, one hasdet(S _(2N))=det(S ₂₂S₁₂ ⁻¹ S ₁₁−S₂₁)det(S ₁₂).  (42)

The matrix s₂₂s₁₂ ⁻¹s₁₁−s₂₁ is thus non-singular as well. By assigning:$\begin{matrix}\left\{ {\begin{matrix}{U_{N} = \left( {{S_{22}S_{12}^{- 1}S_{11}} - S_{21}} \right)^{- 1}} \\{A_{N} = {\left( {{- I_{N}} + S_{22}} \right)S_{12}^{- 1}}} \\{B_{N} = {S_{12}U_{N}^{- 1}}} \\{C_{N} = U_{N}} \\{D_{N} = {{- U_{N}^{- 1}} + {S_{12}^{- 1}S_{11}}}}\end{matrix},} \right. & (43)\end{matrix}$One can easily verify that (39) holds.

Using equation (39), one can derive a reversible transform from thelinear transform of S_(2N). The operation flow of the resultantreversible transform 600 can be shown in FIG. 6. The input 602 of thereversible transform is a size 2N (for real transform) or 4N (for acomplex transform, as each complex consists of an integer real part andan integer imaginary part) integer vectors. After the permutationoperation Q_(2N) 604 it is split into two size N (real) or size 2N(complex) integer vectors x and Y, which are transformed through:$\begin{matrix}\left\{ {\begin{matrix}{Y_{1} = {Y + \left\lbrack {D_{N}X} \right\rbrack}} \\{X_{1} = {X + \left\lbrack {C_{N}Y} \right\rbrack}} \\{X^{\prime} = {{revB}_{N}\left( X_{1} \right)}} \\{Y^{\prime} = {Y_{1} + \left\lbrack {A_{N}X} \right\rbrack}}\end{matrix},} \right. & (44)\end{matrix}$where A_(N) 604, C_(N) 606 and D_(N) 608 are float transforms and [x]represents a vector rounding operation in which the float intermediatevector x is rounded to an integer vector. Under the Cartesiancoordinate, [x] can be implemented via rounding every element of x. Fora complex vector of x, one may individually round the real and imaginarypart of every element of x. Rev B_(N) 612 is a reversible transform tobe derived from the linear non-singular transform B_(N). Finally,another permutation operation P_(2N) 614 is applied on the resultantinteger vectors X′ and Y′. Because each of the above operations can beexactly reversed, the entire transform is reversible.

The inverse of the transform is shown in FIG. 7. In this case, thepermutation operation P_(2N) ^(t) 704 is performed on the input 702.After the permutation operation P₂N^(t) 704, integer vectors X′ and Y′are transformed to integer vectors x and Y through the inverse transformRev B^(-t) _(N) 706, where A_(N) 708, C_(N) 710 and D_(N) 712 are floattransforms. Finally, another permutation operation Q_(2N) ^(t) 714 maybe applied.

Each operation in (44) is called matrix lifting, because it bearssimilarity to the lifting used in (9), except that the multiplicationoperation now is a matrix multiplication, and the rounding operation isa vector rounding. Note that this approach is different from theapproach of [11], where matrix S_(2N) is factored into UTM matrices,each row of which is still calculated via scalar lifting. Matrix liftingwith real-values is used in constructing wavelet [5][7] and DCT-IVtransforms[6]. In the system and method of the invention a matrixlifting of the complex transform is developed, and used a systematic wayin constructing the wavelet transform.

Examining the matrix lifting operation, one notices that it operates ontwo integer vectors of equal size. During the matrix lifting operation,one of the integer vector is kept untouched. This fixed vector also goesthrough a certain float transform with the resultant vector beingrounded. The rounded vector is then added to or subtracted from thesecond vector. The above operation can be reversed by flipping theaddition and subtraction operation. Thus, all matrix lifting operationis reversible.

Using different permutation matrixes P_(2N) and Q_(2N), one may derivedifferent forms of the reversible transforms from the linear transformS_(2N) with different lifting matrixes A_(N), C_(N) and D_(N) and thereversible core B_(N). The trick is to select the permutation matrixesP_(2N) and Q_(2N) SO that:

-   -   a) The reversible core B_(N) is as simple as possible. In the        best case scenarios, the reversible core B_(N) consists of only        permutations. In the sub-optimal case, as in the reversible FFT,        the reversible core B_(N) consists of reversible rotations,        which can be implemented via o(N) lifting steps. If simple        reversible core B_(N) can be found, one may derive a reversible        transform with o(N) rounding operations from a linear transform        of size N. Compared to turning every small module, e.g.,        rotation, into the reversible rotation, which requires O(NlogN)        roundings, the matrix lifting may greatly reduce the number of        rounding operations required in the reversible transform, and        lower the quantization noise of the reversible transform.    -   b) The computation complexity of the transforms A_(N), C_(N) and        D_(N) is as low as possible.        2.3.2 Reversible FFT via Matrix Lifting

The following section derives the reversible FFT with the matrix liftingtool. Inspired by the radix-2 FFT of (29), a 2N-point FFT can befactored into: $\begin{matrix}{{F_{2N} = {\begin{bmatrix}{\frac{1}{\sqrt{2}}F_{N}} & {\frac{1}{\sqrt{2}}{\Lambda_{N}\left( {0.5,0} \right)}F_{N}} \\{\frac{1}{\sqrt{2}}F_{N}} & {{- \frac{1}{\sqrt{2}}}{\Lambda_{N}\left( {0.5,0} \right)}F_{N}}\end{bmatrix}{OE}_{2N}}},} & (45)\end{matrix}$

Setting P_(2N)=I_(2N), Q_(2N)=OE_(2N) and using (39), matrix F_(2N) canbe factorized into the matrix lifting form of (39), with:$\begin{matrix}\left\{ \begin{matrix}{{A_{N} = {{{- \sqrt{2}}F_{N}^{T}{\Lambda_{N}\left( {{- 0.5},0} \right)}} - I_{N}}},} \\{{B_{N} = {{{- {\Lambda_{N}\left( {0.5,0} \right)}}F_{N}F_{N}} = {{- {\Lambda_{N}\left( {0.5,0} \right)}}T_{N}}}},} \\{{C_{N} = {{- \frac{1}{\sqrt{2}}}F_{N}^{T}}},} \\{D_{N} = {\left( {{\sqrt{2}I_{N}} + {F_{N}^{T}{\Lambda_{N}\left( {{- 0.5},0} \right)}}} \right){F_{N}.}}}\end{matrix} \right. & (46)\end{matrix}$

In (46), F_(N) ^(t) is the inverse FFT, T_(N) is a permutation matrix inthe form of: $\begin{matrix}{T_{N} = {\begin{bmatrix}1 & \quad & \quad & \quad \\\quad & \quad & \quad & 1 \\\quad & \quad & \ddots & \quad \\\quad & 1 & \quad & \quad\end{bmatrix}.}} & (47)\end{matrix}$

The reversible core B_(N) is a permutation T_(N) followed by N rotationsΛ_(N)(0.5,0), which can be implemented via the multiple factorizationreversible rotation developed in Section 2.2. The float transform A_(N)consists of an inverse FFT, N rotations, and a vector addition (Scale by2 can be rolled into either the FFT or the rotation operationsΛ_(N)(α,β), with no additional complexity required). Transform C_(N) isan inverse FFT. The transform D_(N) consists of a forward FFT, aninverse FFT and N rotations. An N-point reversible FFT (with 2N inputintegers, as each input is complex with real and imaginary parts) canthus be implemented via (46), with the computation complexity being fourN/2-point complex FFT, N float rotations, and N/2 reversible rotations.It requires 4.5N roundings, with N roundings being used after eachmatrix lifting A_(N), C_(N) and D_(N), and 1.5N roundings being used inthe N/2 reversible rotations in B_(N). Comparing with using reversiblerotation to directly implement the reversible FFT, which requiresO(NlogN) roundings, the matrix lifting approach greatly reduces thenumber of rounding operations.

2.3.3 Reversible Fractional-Shifted FFT via the Matrix Lifting

One may implement the RMDCT with the reversible FFT developed above.Yet, there is even simpler implementation of the RMDCT. Observing thatthe type-IV DST, which is the most important component of the FMDCT, isdirectly related with the fractional shifted FFT F_(N)(α,β) withα=β=0.25 through (16), one may derive a reversible fractional-shiftedFFT directly with matrix lifting. One notices that the fractionalshifted FFT with α=β has following properties:R _(N)(α)=F _(N)(α,α)Λ_(N)(−2α,α)F _(N)(α,α),  (48)where R_(N)(α) is a permutation matrix with only the element (0,0) andelements (i, N-i), i=1, . . . , N-1 being non-zero.

The proof is rather straight forward. LetR_(N)(α)=[r_(N)(i,k)]_(i, k=0, 1, . . . , N-1), the element of R_(N)(α)may be calculated through matrix rotation as: $\begin{matrix}\begin{matrix}{{r_{N}\left( {i,k} \right)} = {\frac{1}{N}{\sum\limits_{j = 0}^{N - 1}{W_{N}^{{({i + \alpha})}{({j + \alpha})}}W_{N}^{{({{- 2}\quad\alpha})}{({j + \alpha})}}W_{N}^{{({j + \alpha})}{({k + \alpha})}}}}}} \\{= {\frac{1}{N}{\sum\limits_{j = 0}^{N - 1}W_{N}^{{({i + k})}{({j + \alpha})}}}}} \\{= {W_{N}^{{({i + k})}\alpha}{{\delta\left( {\left( {i + k} \right){mod}\quad N} \right)}.}}}\end{matrix} & (49)\end{matrix}$

Thus, one has: $\begin{matrix}{{R_{N}(\alpha)} = \begin{bmatrix}1 & \quad & \quad & \quad \\\quad & \quad & \quad & W_{1}^{\alpha} \\\quad & \quad & \ddots & \quad \\\quad & W_{1}^{\alpha} & \quad & \quad\end{bmatrix}} & (50)\end{matrix}$

To derive the reversible transform from the fractional-shifted FFT withα=β=0.25, one again uses the radix-2 FFT structure. Noticing (49), onefactors the fractional shifted FFT as follows:F _(2n)(α,β)=K _(2N)S_(2N)OE_(2N),  (51)with: $\begin{matrix}{{K_{2N} = \begin{bmatrix}{\Lambda_{N}\left( {{{\left( {1 + \beta} \right)/2} - \alpha},\alpha} \right)} & \quad \\\quad & {W_{2}^{\beta}{\Lambda_{N}\left( {{{\left( {1 + \beta} \right)/2} - \alpha},\alpha} \right)}}\end{bmatrix}},} & (52) \\{{{and}\quad S_{2N}} = {\begin{bmatrix}{\frac{1}{\sqrt{2}}{\Lambda_{N}\left( {{- \frac{1}{2}},\alpha} \right)}{F_{N}\left( {\alpha,\alpha} \right)}} & {\frac{1}{\sqrt{2}}{F_{N}\left( {\alpha,\alpha} \right)}} \\{\frac{1}{\sqrt{2}}{\Lambda_{N}\left( {{- \frac{1}{2}},\alpha} \right)}{F_{N}\left( {\alpha,\alpha} \right)}} & {{- \frac{1}{\sqrt{2}}}{F_{N}\left( {\alpha,\alpha} \right)}}\end{bmatrix}.}} & (53)\end{matrix}$

Substitute α=0.25 and expanding the fractional-shifted FFT via (23), onefactors the transform S_(2N) into the matrix lifting form of (39), with:$\begin{matrix}{\quad\left\{ \begin{matrix}{{A_{N} = {{{- \sqrt{2}}{\Lambda_{N}\left( {{- 0.25},0.25} \right)}F_{N}^{T}{\Lambda_{N}\left( {{- 0.25},0} \right)}} - I_{N}}},} \\{{B_{N} = {{- {R_{N}(0.25)}} = \begin{bmatrix}{- 1} & \quad & \quad & \quad \\\quad & \quad & \quad & j \\\quad & \quad & \ddots & \quad \\\quad & j & \quad & \quad\end{bmatrix}}},} \\{{C_{N} = {{- \frac{1}{\sqrt{2}}}{\Lambda_{N}\left( {{- 0.25},0.25} \right)}F_{N}^{T}{\Lambda_{N}\left( {0.25,0.5} \right)}}},} \\{D_{N} = {{\Lambda_{N}\left( {{- 0.25},0.25} \right)}\left( {\sqrt{2} + {F_{N}^{T}{\Lambda_{N}\left( {{- 0.5},0.125} \right)}}} \right)F_{N}{{\Lambda_{N}\left( {0.25,0} \right)}.}}}\end{matrix} \right.} & (54)\end{matrix}$

In (54), the reversible core B_(N) is simply permutation, as multiplyingby j just swaps the real and imaginary part and changes the sign of theimaginary part. The reversible fractional-shifted FFT of α=β=0.25 isthus the matrix lifting of (54) plus the post reversible rotations ofK_(2N), which again can be implemented via the multiple factorizationrotations described in Section 2.3. Using (54), an N-point RMDCT can beimplemented via N/2 reversible window rotations of h(n), a reversiblematrix lifting of S_(N/2), and N/2 reversible rotations of K_(N/2)(noticing that the N-point FMDCT consists of an N-point type-IV DST,which can be further converted to an N/2-point fractional-shifted FFT).The total computational complexity is N reversible rotations, fourN/4-point float FFTs, and 1.75N float rotations. The implementationcomplexity is about double of an FMDCT, which requires two N/4-pointFFTs and 1.5N float rotations. Altogether, the RMDCT requires 4.5Nroundings, with three 0.5N roundings after each matrix lifting of (54),and 3N roundings for the N reversible rotations.

2.4 Reversible MDCT: the Integer Arithmetic

Most operations of the reversible transform, e.g., the add/subtractoperation in the lifting, the permutation, and the sign changeoperation, are integer operations. The only place that requires thefloat operation is in the lifting, where the input integer value(vector) is multiplied by a float value (or transformed through a floatmatrix), and then rounded. The lifting operation can certainly beimplemented via the float arithmetic, e.g., with double precision, whichprovides high precision and large dynamic range. However, the floatarithmetic is inconsistent across machines, and therefore, reversibilitycan not be guaranteed across platforms. Moreover, the float arithmeticis also more complex. Since the float result is ultimately rounded afterthe lifting, high precision float operation is not essential in thereversible transform. The float operation of the reversible transformmay thus be implemented with the integer arithmetic by representing boththe input, result and the transform coefficients with fixed-pointinteger, which is represented in the computer as a single integer.

To implement the lifting operation with the integer arithmetic, eachoperand of the float operation is interpreted as a fixed precision floatnumber:±b_(n)b_(n-1) . . . b₁.a₁a₂ . . . a_(m),  (55)where m is the number of bits of the fractional part, and n is thenumber of bits of the integer part. The representation in (55) requiresa total of n+m+1 bits (with one bit for sign). The variable n controlsthe dynamic range, and m controls the amount of quantization noiseintroduced by the fixed-point arithmetic. A fixed precision float in(55) may represent values with absolute magnitude up to 2^(n)-2^(-m),and with precision down to 2^(-m). To perform a float operation:y=w·x,  (56)where x is the input, y is the output, and w is the multiplicationvalue, the following operations are performed. Assuming that the inputand output x and y are represented with n_(X) bit dynamic range andm_(x) bit precision, and the transform coefficient/multiplication valuew is represented with n_(w) bit dynamic range and m_(w) bit precision,one may treat x, y and w as integer values, perform the multiplication,and then round the result by m_(w) bits.

The operations of using fixed-precision float to implement the matrixlifting can be shown in FIG. 8. First, the integer input vector isconverted to the fixed-point float representation (process actions 810,820) by left shifting the input vector by a fixed number of bits (m bitsin above). Next, a fixed-precision float transform is performed on theinput using stored fixed-point float coefficients (process actions 830,840). The stored fixed-point float coefficients are the parameters ofthe matrix lifting, e.g., the content of matrix A_(N), C_(N), and D_(N)in equation (44). Finally, the output fixed-point result is rounded tointeger, as shown in process action 850. The rounding operation isperformed by right shifting the resultant vector by m bits andcompensating the carryover bits. The assemble code to perform therounding operation can be shown as: { SHR [x], m; ADC [x], 0; }

Where [x] is the fixed-precision float that is to be rounded.

The key of the fixed-point implementing of the matrix lifting is to makesure that the intermediate result does not overflow, i.e., goes out ofthe dynamic range allowed by the fixed-point float representation, andto keep the additional quantization noise caused by the fixed-pointarithmetic negligible compared to the quantization noise of the roundingoperation.

The system and method of the invention also employs a special procedureto reduce the dynamic range required for the FFT transform used in thematrix lifting. Traditional implementation of a normalized FFT can beshown in FIG. 9 or FIG. 10. Referring to FIG. 9, after the signal isinput (process action 910), 2 ^(N) point FFT is implemented through Nstages of a butterfly operation of equation (30) (process actions 920,930, 940). A scaling by dividing 2^(N/2) operation is performed eitherafter (in FIG. 9) or before (in FIG. 10, process action 1020) the groupof butterfly operations. However, performing the scaling operation afterthe group of butterfly operations, as in FIG. 9, requires an additionalN/2 bits to accommodate the increase of the dynamic range during the FFToperation. Performing the scaling operation before the group ofbutterfly operations, as in FIG. 10, increases the quantization noise ofthe fixed-point FFT implementation, and is also unfavorable. In thepresent invention, the FFT implementation shown in FIG. 11 was adopted.A division by 2 operation (process action 1140) was inserted after eachtwo butterfly operations (process actions 1120, 1130). A final divisionby 2 ^(1/2) operation is performed if the number of butterfly stages Nis odd (process action 1160). In this way, the precision of thefixed-point FFT implementation is maximized while not increasing thedynamic range of the implementation. The final division by 2^(1/2)operation is usually combined with other float transforms, so it usuallydoes not need a separate processing step.

The only component left to find is the required dynamic range and bitprecision of the input, the output, and the transform coefficient. Theseparameters may be derived empirically. First, the dynamic range and bitprecision needed to represent the transform coefficient areinvestigated, which in RMDCT, is mainly the rotation angle W_(N) ^(i).The rotation can be implemented either via a 2×2 matrix multiplication,where the values cos θand sin θare used, or via a 3-step multiplefactorization lifting developed in Section 2.3, where the values c₀=(cosθ-1)/sin θ and c₁=sin θ are used. In the multiple factorizationreversible rotation, the absolute value of C₀ can reach 2.414, whichneeds n_(w)=2 bit dynamic range. Thus, if the transform coefficient isrepresented with a 32 bit integer, it can have a bit precision of atmost m_(w)=29 bits.

To illustrate the impact of the bit precision of the transformcoefficient on the quantization noise level of the reversible transform,the magnitude of the quantization noise of the RMDCT versus that of theFMDCT is measured, under different bit precisions of the transformcoefficients. The quantization noise is measured in terms of the meansquare error (MSE), the mean absolute difference (MAD) and the peakabsolute difference (PAD), where $\begin{matrix}{{{MSE} = {\frac{1}{N}{\sum\limits_{i = 0}^{N - 1}\left( {y_{i}^{\prime} - y_{i}} \right)^{2}}}},} & (57) \\{{{MAD} = {\frac{1}{N}{\sum\limits_{i = 0}^{N - 1}{{y_{i}^{\prime} - y_{i}}}}}},} & (58) \\{{{and}\quad{PAD}} = {\max\limits_{i}{{{y_{i}^{\prime} - y_{i}}}.}}} & (59)\end{matrix}$

In the above, y_(i) is the FMDCT coefficient, and y′_(i) is the RMDCTcoefficient. The result can be shown in Table 4. The RMDCT in use isderived via the fractional-shifted FFT of Section 2.3.3. In the secondcolumn of Table 4 the quantization noise level of the RMDCT implementedvia the float arithmetic is shown. Then, in the following columns thequantization noise level of the RMDCT implemented via the integerarithmetic is shown, with the bit precision of the transformcoefficients m_(w) being 29, 20, 16 and 12 bits. It is observed thatwith a bit precision above 16 bits, the RMDCT implemented via theinteger arithmetic has a quantization noise level very close to that ofthe float arithmetic. Less bit precision significantly increases thequantization noise level of the reversible transform, as there is notenough accuracy to correctly represent the multiplicativevalue/transform coefficient. The bit precision for the transformcoefficient m_(w) is chosen to be 29 bits, as this still allows thetransform coefficient to be represented with 32 bit integer. For theother three bits, two bits are used for the dynamic range of thetransform coefficients (n_(w)=2), and one bit is used for the sign.TABLE 1 Bit precision of the transform coefficient m_(w) Float Precisionarithmetic m_(w) = 29 20 16 12 MSE 0.47 0.47 0.48 0.49 5.94 MAD 0.530.53 0.54 0.54 1.18 PAD 11.86 11.86 11.86 11.86 286.56

Next, the bit precision m_(x) required to represent the input and outputof the matrix lifting operations is investigated. Again, thequantization noise level of the RMDCT versus that of the FMDCT iscompared, with different bit precisions of the input and output. Theresult is shown in Table 2. It is evident that the quantization noiselevel starts to increase with less than m_(x)=5 bits to represent theintermediate result of the float transform. TABLE 2 Bit precision of theinput/output of the matrix lifting m_(x). Preci- float sion arithmeticm_(x) = 9 5 4 3 2 1 0 MSE 0.47 0.47 0.48 0.51 0.60 0.98 2.47 8.08 MAD0.53 0.53 0.54 0.55 0.58 0.69 0.98 1.73 PAD 11.86 11.86 11.86 10.8610.02 15.17 27.38 49.16

Finally, the dynamic range n_(x) needed to represent the input andoutput of the matrix lifting is investigated. It is noticed that alloperations used in the RMDCT, whether the float rotation, the reversiblerotation, or the FFT, are energy preserving operations. Thus, themaximum magnitude that a coefficient can reach is:max=2 ^(bitdepth-1){square root}{square root over (2N)},  (60)where bitdepth is the number of bits of the input audio, and N is thesize of the MDCT transform. One bit (noticing {square root}{square rootover (2)} in (54)) is needed to further guard against overflow in theRMDCT. The dynamic range n_(X) needed for the matrix lifting is thus:$\begin{matrix}{{n_{x} = {{bitdepth} + {\frac{1}{2}{\log_{2}\left( {2N} \right)}}}},} & (61)\end{matrix}$

Table 3 Maximum bit depth of the input audio that can be fed into anRMDCT with 32 bit integer. Precision m_(x) used 9 8 7 6 5 4 N = 256 1718 19 20 21 22  N = 1024 16 17 18 19 20 21  N = 4096 15 16 17 18 19 22

In Table 3, the maximum bitdepth of the input audio that can be fed intothe RMDCT with 32 bit integer arithmetic implementation is listed, withdifferent bit precision m_(x) and RMDCT size. This table has beenverified by feeding a pure sine wave of maximum magnitude into the RMDCTmodule, and by making sure that there is no overflow in the RMDCTmodule. With the RMDCT block size N being 1024, the RMDCT with 32 bitinteger arithmetic may accommodate a 20-bitdepth input audio withm_(x)=5 bits left to represent the fractional part of the lifting.

2.5 Exemplary Progressive-to-lossless Audio Codec (PLEAC) with RMDCT.

Using a RMDCT and a lossless embedded entropy coder, such as for examplethe one developed in [13], a progressive to lossless embedded audiocoder (PLEAC) is developed in accordance with the system and method ofthe present invention. The encoder framework of PLEAC is shown in FIG.12.

If the input audio 1202 is stereo, the audio waveform first goes througha reversible multiplexer (MUX) 1204, whose formulation can be shown withequation (2), and separates into the L+R and L-R components, where L andR represent the audio on the left and right channel, respectively. Ifthe input audio is mono, the MUX simply passes through the audio. Thewaveform of each audio component is then transformed by an RMDCT module1206 a, 1206 b with switching windows. The RMDCT window size switchedbetween 2048 and 256 samples. After the RMDCT transform, the RMDCTcoefficients of a number of consecutive windows are grouped into atimeslot. In the current configuration, a timeslot consists of 16 longwindows or 128 short windows, which in either case include 32,768samples. The coefficients in the timeslot are then entropy encoded by ahighly efficient psychoacoustic embedded entropy coder 1208 a, 1208 b.The entropy encoder 1208 a, 1208 b generates a bitstream that ifdelivered in its entity to the decoder, may losslessly decode the inputcoefficients. Yet, the bitstream can be truncated at any point withgraceful psychoacoustic quality degradation. Finally, a bitstreamassembly 1210 module puts the bitstreams of both channels together, andforms the final compressed bitstream. A conventional sub-bitplaneentropy coder and conventional bitstream assembly module may be used.

The framework of the PLEAC decoder can be shown in FIG. 13. The receivedbitstream 1302 is first split into individual channel bitstreams by adisassembly module 1304. Then, it is decoded by the embedded entropydecoder 1306 a, 1306 b. If the decoder 1306 a, 1306 b finds that thereceived bitstream is lossless or close to lossless, i.e., thecoefficients can be decoded to the last bitplane, the decodedcoefficients are transformed by an inverse RMDCT module 1308 a, 1308 b.Otherwise, an inverse FMDCT module 1310 a, 1310 b is used. The rationalto apply the inverse FMDCT for the lossy bitstream is that the inverseFMDCT not only has lower computational complexity, but also generates noadditional quantization noise. In comparison, the inverse RMDCTgenerates quantization noise, which when decoded to lossless, serves tocancel the quantization noise of the encoding stage, but when decoded tolossy is just additional noise, which degrades the audio playbackquality. After the inverse MDCT transform, the individual audio channelsare then demultiplexed by a multiplexer 1312 to form the decoded audiowaveform 1314.

Notice that the RMDCT module is used in the PLEAC encoder, whereas inthe PLEAC decoder it is only used in lossless decoding. Thecomputational penalty of RMDCT thus only resides with the PLEAC encoderand lossless PLEAC decoder.

2.6 Exemplary Methods for Implementing a RMDCT via Matrix Lifting

Turning briefly to FIGS. 14 and 15, methodologies that may beimplemented in accordance with the present invention are illustrated.While, for purposes of simplicity of explanation, the methodologies areshown and described as a series of blocks, it is to be understood andappreciated that the present invention is not limited by the order ofthe blocks, as some blocks may, in accordance with the presentinvention, occur in different orders and/or concurrently with otherblocks from that shown and described herein. Moreover, not allillustrated blocks may be required to implement the methodologies inaccordance with the present invention.

The invention may be described in the general context ofcomputer-executable instructions, such as program modules, executed byone or more components. Generally, program modules include routines,programs, objects, data structures, and so forth, that performparticular tasks or implement particular abstract data types. Typicallythe functionality of the program modules may be combined or distributedas desired in various embodiments.

Referring to FIG. 14, an exemplary method for lossless data orprogressive to lossless encoding 1400 in accordance with an aspect ofthe present invention is illustrated. At 1410, a media input signal isreceived. At 1420, quantized coefficients corresponding to the inputsignal based, at least in part, upon a reversible modified discretecosine transform obtained via matrix lifting is provided. For example,the reversible modified discrete cosine transform can be based upon aFFT or a fractional-shifted FFT. At 1430, the quantized coefficients areentropy encoded into an embedded bitstream, which may be truncated orreshaped later.

Next, turning to FIG. 15, an exemplary method for lossless data decoding1500 in accordance with an aspect of the present invention isillustrated. At 1510, a digitally entropy encoded input bit stream isreceived. At 1520, the bit stream is entropy decoded and transformcoefficients are provided. At 1530, it is tested whether the inputbitstream is lossless or close to lossless. If the test is true, theoutput values based on an inverse reversible transform 1550 of thetransform coefficients are provided. If the test is not true, the outputvalues based on an inverse float (linear) transform 1540 of thetransform coefficients are provided.

3.0 EXPERIMENTAL RESULTS

To evaluate the performance of the proposed RMDCT, the RMDCT moduleswere put into the progressive to lossless embedded audio codec (PLEAC).Then the following RMDCT configurations were compared:

-   -   (a) Through the reversible fractional-shifted FFT via the matrix        lifting described in Section 2.4.    -   (b) Same as (a), except that the reversible rotation is        implemented with only the factorization form of (8).    -   (c) Same as (a), except that the rounding operation is        implemented as truncation towards zero.    -   (d) Through the reversible FFT via the matrix lifting described        in Section 2.4.        With only multiple factorization reversible rotations described        in Section 2.3.

All the other modules of the PLEAC codec are the same. First the outputdifference of the RMDCT modules versus that of the FMDCT module werecompared, in terms of the mean square error (MSE), the mean absolutedifference (MAD) and the peak absolute difference (PAD) calculated inequations (57)-(59). Then the lossless compression ratio of the PLEACcodecs using the specific RMDCT configuration with the state-of-the-artlossless audio compressor, the Monkey's Audio [11] were compared.Finally, the lossy coding performance was compared. The lossy compressedbitstream was derived by truncating the losslessly compressed bitstreamto the bitrate of 64, 32 and 16 kbps. Then the lossy bitstream wasdecoded, and the decoding noise-mask-ratio (NMR) versus that of theoriginal audio waveform (the smaller the NMR, the better the quality ofthe decoded audio) was measured. The NMR results were then compared withthose of the lossy EAC codec, which is the PLEAC with the FMDCT modulein both the encoder and the decoder. The test audio waveform was formedby concatenating the MPEG-4 sound quality assessment materials (SQAM).The aggregated comparison results are shown in Table 4 and Table 5.TABLE 4 Quantization noise levels of different RMDCT modules. PLEAC w/RMDCT versus FMDCT a b c d e MSE 0.48 2.18 3.03 0.81 1.78 MAD 0.54 0.681.14 0.69 1.04 PAD 11.86 466.48 34.22 11.10 18.32

TABLE 5 Lossless and lossy compression performance of PLEAC. Monkey'sEAC w/ PLEAC w/ PLEAC w/ PLEAC w/ PLEAC w/ PLEAC w/ Audio FMDCT Audiocodec RMDCT a RMDCT b RMDCT c RMDCT d RMDCT e (lossless) (lossy)Lossless 2.88:1 2.73:1 2.85:1 2.85:1 2.77:1 2.93:1 compression ratioLossy NMR −2.18 4.13 −1.60 −1.73 −0.06 −3.80 (64 kbps) Lossy NMR 2.266.69 2.64 2.48 3.63 1.54 (32 kbps) Lossy NMR 5.41 8.23 5.63 5.49 6.115.37 (16 kbps)

Because the configuration (b) only differs from the configuration (a) inthe implementation of the reversible rotations, their difference inTable 4 demonstrates the effectiveness of the multiple factorizationreversible rotations. By intelligently selecting the properfactorization form under different rotation angles, multiplefactorization greatly reduces the quantization noise in the rotations.The MSE of the quantization noise is reduced by 78%. There is also anoticeable improvement in the lossless (5% less bitrate) and lossy (onaverage 4.5 dB better) coding performance of the PLEAC codec byreplacing reversible rotation of (8) with the multiple factorizationreversible rotations.

The configuration (c) only differs from the configuration (a) in theimplementation of the rounding operation, with configuration (a) usesrounding towards the nearest integer, while configuration (c) uses therounding towards zero. Though the two schemes only differ slightly inimplementation, rounding towards the nearest integer is proven to be abetter choice for rounding. As shown in Table 4 and 5, configuration (a)reduces the MSE by 84%, with slightly better lossless (1% less bitrate)and lossy (on average 0.4 dB better) compression performance.

From configuration (e) to (d) to (a), the matrix lifting is used toimplement an ever-larger chunk of the FMDCT into a reversible module.Comparing configuration (e) (only reversible rotations) withconfiguration (a), which uses the matrix lifting on thefractional-shifted FFT, the quantization noise of the RMSE is reduced by73%, while there is a reduction of 4% of lossless coding bitrate. Onealso observes that the NMR of lossy decoded audio improves by an averageof 1.4 dB.

Overall, the RMDCT configuration with lower quantization noise leads tobetter lossless and lossy audio compression performance. The anomalylies in the configuration (c). Though using truncation towards zeroresults in a big increase in the quantization noise in term of MSE, MADand PAD, it does not incur as much penalty in the lossless and lossycompression performance as compared with configurations (b) and (e).This anomaly may be explained by the fact that truncation towards zerogenerally results in smaller entropy of the RMDCT coefficients, whichmitigates some of the entropy increase caused by the rising quantizationnoise. Nevertheless, it is noticed that rounding towards the nearestinteger still leads to superior performance in lossless and lossycompression.

It is observed that with the matrix lifting, the output of the RMDCTbecomes very close to the FMDCT. The best RMDCT configuration (a), whichis implemented via the reversible fractional-shifted FFT with the matrixlifting and the multiple-factorization reversible rotations, results inthe MSE of the quantization noise of only 0.48. Therefore, a largenumber of RMDCT coefficients are just the same as the FMDCT coefficientsafter rounding. Incorporating the RMDCT module (a), the PLEAC codecachieves a lossless compression ratio of 2.88:1, while thestate-of-the-art lossless audio compressor, the Monkey's Audio[11],achieves a lossless compression ratio of 2.93:1 of the same audiowaveform. PLEAC with RMDCT is thus within 2% of the state-of-the-artlossless audio codec. Moreover, the PLEAC compressed bitstream can bescaled, from lossless all the way to very low bitrate, whereas suchfeature is non-existing in the Monkey's Audio. Comparing with the EACcodec, which uses FMDCT in both the encoder and decoder, PLEAC onlyresults in an NMR loss of 0.8 dB. PLEAC is thus a fantastic all-aroundscalable codec from lossy all the way to lossless.

The foregoing description of the invention has been presented for thepurposes of illustration and description. It is not intended to beexhaustive or to limit the invention to the precise form disclosed. Manymodifications and variations are possible in light of the aboveteaching. It is intended that the scope of the invention be limited notby this detailed description, but rather by the claims appended hereto.

REFERENCES

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1. A system for encoding a media signal comprising: inputting a signalin the form of an integer vector; and using a reversible transformobtained with at least one reversible matrix lifting operation to encodesaid signal, comprising: splitting the integer vector into two complexinteger vectors of equal size; keeping one of the two complex integervectors of equal size fixed; transforming the fixed complex integervector via a float transform to obtain a transformed vector; roundingthe transformed vector to obtain a resultant vector; either adding theresultant vector to, or subtracting the resultant vector from, the oneof the two complex integer vectors that was not kept fixed therebygenerating an encoded version of the input signal in integer form. 2.The system of claim 1 wherein the reversible transform is in the form ofa reversible modified discrete cosine transform.
 3. The system of claim1 wherein the reversible transform has a 2N input and a 2N output with anon-singular characterization matrix S_(2N).
 4. The system of claim 3wherein the reversible transform comprises:${S_{2N} = {{{{{P_{2N}\begin{bmatrix}I_{N} & \quad \\A_{N} & I_{N}\end{bmatrix}}\begin{bmatrix}B_{N} & \quad \\\quad & I_{N}\end{bmatrix}}\begin{bmatrix}I_{N} & C_{N} \\\quad & I_{N}\end{bmatrix}}\begin{bmatrix}I_{N} & \quad \\D_{N} & I_{N}\end{bmatrix}}Q_{2N}}},$ where P_(2N) and Q_(2N) are permutationoperations, and wherein after permutation P_(2N), the input signal isdivided into two integer vectors X and Y, which are transformed toresultant integer vectors X′ and Y′ through: $\left\{ {\begin{matrix}{Y_{1} = {Y + \left\lbrack {D_{N}X} \right\rbrack}} \\{X_{1} = {X + \left\lbrack {C_{N}Y} \right\rbrack}} \\{X^{\prime} = {{revB}_{N}\left( X_{1} \right)}} \\{Y^{\prime} = {Y_{1} + \left\lbrack {A_{N}X} \right\rbrack}}\end{matrix},} \right.$ where A_(N), C_(N) and D_(N) are floattransforms, [D_(N)X] [C_(N)Y] [A_(N)X] represent vector roundingoperations, and Rev B_(N) is a reversible transform derived fromreversible transform B_(N).
 5. The system of claim 4 wherein vectorrounding comprises rounding the individual component of each vector. 6.The system of claim 5 wherein rounding the individual componentcomprises separately rounding the real and complex coefficients of theindividual component.
 7. The system of claim 4 wherein different formsof the reversible transform can be obtained from the linear transformS_(2N) using different permutation matrixes P_(2N) and Q_(2N), differentfloat transforms A_(N), C_(N) and D_(N), and a different reversibletransform B_(N).
 8. The system of claim 1 wherein the input signalcomprises an audio signal.
 9. The system of claim 1 wherein the inputsignal comprises an image signal.
 10. The system of claim 1 wherein thereversible transform is a reversible Fast Fourier Transform (FFT). 11.The system of claim 1 wherein the reversible transform is a reversiblefractional-shifted Fast Fourier Transform (FFT).
 12. The system of claim8 wherein the reversible Fast Fourier Transform is a 2N-point reversibleFast Fourier Transform (FFT) F_(2N) implemented via:${F_{2N} = {{{{{P_{2N}\begin{bmatrix}I_{N} & \quad \\A_{N} & I_{N}\end{bmatrix}}\begin{bmatrix}B_{N} & \quad \\\quad & I_{N}\end{bmatrix}}\begin{bmatrix}I_{N} & C_{N} \\\quad & I_{N}\end{bmatrix}}\begin{bmatrix}I_{N} & \quad \\D_{N} & I_{N}\end{bmatrix}}Q_{2N}}},$ wherein P_(2N)=I_(2N), Q_(2N)=OE_(2N) and theother matrixes in the matrix lifting are: $\quad\left\{ \begin{matrix}{{A_{N} = {{{- \sqrt{2}}F_{N}^{T}{\Lambda_{N}\left( {{- 0.5},0} \right)}} - I_{N}}},} \\{{B_{N} = {{{- {\Lambda_{N}\left( {0.5,0} \right)}}F_{N}F_{N}} = {{- {\Lambda_{N}\left( {0.5,0} \right)}}T_{N}}}},} \\{{C_{N} = {{- \frac{1}{\sqrt{2}}}F_{N}^{T}}},} \\{D_{N} = {\left( {{\sqrt{2}I_{N}} + {F_{N}^{T}{\Lambda_{N}\left( {{- 0.5},0} \right)}}} \right){F_{N}.}}}\end{matrix} \right.$ where F_(N) is the forward FFT, F_(N) ^(t) is theinverse FFT and T_(N) is a permutation matrix in the form of:$T_{N} = {\begin{bmatrix}1 & \quad & \quad & \quad \\\quad & \quad & \quad & 1 \\\quad & \quad & \ddots & \quad \\\quad & 1 & \quad & \quad\end{bmatrix}.}$
 13. The system of claim 11 wherein the reversiblefractional-shifted transform is a 2N-point reversible fractional-shiftedFast Fourier Transform (FFT) implemented via a reversible transformS_(2N) of matrix lifting followed by N reversible rotations K_(2N). 14.The system of claim 13 wherein in the fractionally shiftedFFTF_(2N)(α,β), with α=β=0.25, the reversible rotations K_(2N) take theform ${K_{2N} = \begin{bmatrix}{\Lambda_{N}\left( {{{\left( {1 + \beta} \right)/2} - \alpha},\alpha} \right)} & \quad \\\quad & {W_{2}^{\beta}{\Lambda_{N}\left( {{{\left( {1 + \beta} \right)/2} - \alpha},\alpha} \right)}}\end{bmatrix}},$ and the reversible transform S_(2N) of matrix liftingis implemented via: ${S_{2N} = {{{{{P_{2N}\begin{bmatrix}I_{N} & \quad \\A_{N} & I_{N}\end{bmatrix}}\begin{bmatrix}B_{N} & \quad \\\quad & I_{N}\end{bmatrix}}\begin{bmatrix}I_{N} & C_{N} \\\quad & I_{N}\end{bmatrix}}\begin{bmatrix}I_{N} & \quad \\D_{N} & I_{N}\end{bmatrix}}Q_{2N}}},$ wherein P_(2N)=I_(2N), Q_(2N)=OE_(2N) and thematrixes in the matrix lifting are: $\quad\left\{ \begin{matrix}{{A_{N} = {{{- \sqrt{2}}{\Lambda_{N}\left( {{- 0.25},0.25} \right)}F_{N}^{T}{\Lambda_{N}\left( {{- 0.25},0} \right)}} - I_{N}}},} \\{{B_{N} = {{- {R_{N}(0.25)}} = \begin{bmatrix}{- 1} & \quad & \quad & \quad \\\quad & \quad & \quad & j \\\quad & \quad & \ddots & \quad \\\quad & j & \quad & \quad\end{bmatrix}}},} \\{{C_{N} = {{- \frac{1}{\sqrt{2}}}{\Lambda_{N}\left( {{- 0.25},0.25} \right)}F_{N}^{T}{\Lambda_{N}\left( {0.25,0.5} \right)}}},} \\{D_{N} = {{\Lambda_{N}\left( {{- 0.25},0.25} \right)}\left( {\sqrt{2} + {F_{N}^{T}{\Lambda_{N}\left( {{- 0.5},0.125} \right)}}} \right)F_{N}{{\Lambda_{N}\left( {0.25,0} \right)}.}}}\end{matrix} \right.$
 15. The system of claim 1 wherein the input signalis encoded in a lossless manner.
 16. The system of claim 1 wherein theinput signal is encoded in a progressive-to-lossless manner.
 17. Acomputer-implemented process for compressing an audio signal comprisingthe process actions of, inputting an audio signal; if the input audiosignal is stereo, processing the audio signal through a reversiblemultiplexer (MUX) which separates into the L+R and L−R components, whereL and R represent the audio on the left and right channel, respectively;if the input audio is mono, passing the audio signal through the MUX;transforming the waveform of each audio component by a RMDCT module;grouping the RMDCT coefficients of a number of consecutive windows intoa timeslot; entropy encoding the coefficients in the timeslot using anembedded entropy coder; and putting the bitstreams of the left and rightchannels and timeslots together via a bitstream assembly module to forma final compressed bitstream.
 18. The process of claim 17 wherein theRMDCT module is implemented using switching windows.
 19. Aprogressive-to-lossless audio decoder system comprising: a bit streamunassembler that unassembles the final compressed bitstream intobitstreams of individual channels and timeslots; an embedded entropydecoder that digitally entropy decodes the bitstream of the individualchannels and timeslots; if the input bitstream is lossless or close tolossless, an inverse RMDCT module that transforms the decodedcoefficients to waveform; if the input bitstream is not close tolossless, an inverse FMDCT module that transforms the decodedcoefficients to waveform; if the audio signal is stereo, an inversemultiplexer that combines the L+R and L−R components.
 20. The decoder ofclaim 19 wherein the input bitstream comprises the compressed bits of aseries of bitplanes and wherein the bitstream is substantially losslessif the transform coefficients can be decoded in the last bitplane.
 21. Acomputer-implemented process for encoding media data, comprising theprocess actions of: using a reversible transform component that receivesan input signal and provides an output of quantized coefficientscorresponding to the input signal, the output of quantized coefficientsbeing based, at least in part, upon a reversible transform obtained viamatrix lifting, wherein matrix lifting comprises inputting the signal inthe form of an integer vector; splitting the integer vector into twocomplex integer vectors of equal size; keeping one of the two complexinteger vectors of equal size fixed; transforming the fixed complexinteger vector via a float transform to obtain a transformed vector;rounding the transformed vector to obtain a resultant vector; eitheradding the resultant vector to, or subtracting the resultant vectorfrom, the one of the two complex integer vectors that was not keptfixed; and, using an entropy encoder component that digitally entropyencodes the quantized coefficients.
 22. The computer-implemented processof claim 21 wherein the reversible transform obtained via matrix liftingcomprises at least one float Fast Fourier transform.
 23. Thecomputer-implemented process of claim 21 wherein the Fast FourierTransform is a ²N point FFT implemented by, for 1 to N, performing afirst butterfly operation; performing a second butterfly operation toobtain an interim result; dividing the interim result by 2; and dividingthe interim result by 2^(1/2) operation if the number of butterflystages N is odd.
 24. The computer-implemented process of claim 22wherein the matrix lifting is implemented via fixed floatrepresentation, comprising: converting the input signal to a fixedprecision float representation by left shifting a fixed number of bits;performing a fixed precision float transform using previously-storedfixed precision float coefficients; outputting a fixed precision floatpoint result; and rounding the fixed precision float point result tointeger values by right shifting a fixed number of bits and adding thecarryover bit shifted out.
 25. The computer-implemented process of claim21 wherein said process actions are stored on a computer-readablemedium.
 26. The computer-implemented process of claim 21 wherein thereversible transform is in the form of a reversible modified discretecosine transform.
 27. The computer-implemented process of claim 21wherein the reversible transform has a 2N input and a 2N output with anon-singular characterization matrix S_(2N).
 28. Thecomputer-implemented process of claim 27 wherein the reversibletransform comprises: ${S_{2N} = {{{{{P_{2N}\begin{bmatrix}I_{N} & \quad \\A_{N} & I_{N}\end{bmatrix}}\begin{bmatrix}B_{N} & \quad \\\quad & I_{N}\end{bmatrix}}\begin{bmatrix}I_{N} & C_{N} \\\quad & I_{N}\end{bmatrix}}\begin{bmatrix}I_{N} & \quad \\D_{N} & I_{N}\end{bmatrix}}Q_{2N}}},$ where P_(2N) and Q_(2N) are permutationoperations, and wherein after permutation P_(2N), the input signal isdivided into two integer vectors X and Y, which are transformed toresultant integer vectors X′ and Y′ through: $\left\{ {\begin{matrix}{Y_{1} = {Y + \left\lbrack {D_{N}X} \right\rbrack}} \\{X_{1} = {X + \left\lbrack {C_{N}Y} \right\rbrack}} \\{X^{\prime} = {{revB}_{N}\left( X_{1} \right)}} \\{Y^{\prime} = {Y_{1} + \left\lbrack {A_{N}X} \right\rbrack}}\end{matrix},} \right.$ where A_(N), C_(N) and D_(N) are floattransforms, [D_(N)X] [C_(N)Y] [A_(N)X] represent vector roundingoperations, and Rev B_(N) is a reversible transform derived fromreversible transform B_(N).
 29. The computer-implemented process ofclaim 28 wherein vector rounding is obtained by separately rounding thereal and complex coefficient of the individual component of the vector.30. The computer-implemented process of claim 28 wherein different formsof the reversible transform can be obtained from the linear transformS_(2N) using different permutation matrixes P_(2N) and Q_(2N), differentfloat transforms A_(N), C_(N) and D_(N), and a different reversibletransform B_(N).
 31. The computer-implemented process of claim 21wherein the input signal comprises an audio signal.
 32. Thecomputer-implemented process of claim 21 wherein the input signalcomprises an image signal.
 33. The computer-implemented process of claim21 wherein the reversible transform is a reversible Fast FourierTransform (FFT).
 34. The computer-implemented process of claim 21wherein the reversible transform is a reversible fractional-shifted FastFourier Transform (FFT).
 35. The computer-implemented process of claim33 wherein the reversible Fast Fourier Transform is a 2N-pointreversible Fast Fourier Transform (FFT) F_(2N) implemented via:${F_{2N} = {{{{{P_{2N}\begin{bmatrix}I_{N} & \quad \\A_{N} & I_{N}\end{bmatrix}}\begin{bmatrix}B_{N} & \quad \\\quad & I_{N}\end{bmatrix}}\begin{bmatrix}I_{N} & C_{N} \\\quad & I_{N}\end{bmatrix}}\begin{bmatrix}I_{N} & \quad \\D_{N} & I_{N}\end{bmatrix}}Q_{2N}}},$ wherein P_(2N)=I_(2N), Q_(2N)=OE_(2N) and theother matrixes in the matrix lifting are: $\quad\left\{ \begin{matrix}{{A_{N} = {{{- \sqrt{2}}F_{N}^{T}{\Lambda_{N}\left( {{- 0.5},0} \right)}} - I_{N}}},} \\{{B_{N} = {{{- {\Lambda_{N}\left( {0.5,0} \right)}}F_{N}F_{N}} = {{- {\Lambda_{N}\left( {0.5,0} \right)}}T_{N}}}},} \\{{C_{N} = {{- \frac{1}{\sqrt{2}}}F_{N}^{T}}},} \\{D_{N} = {\left( {{\sqrt{2}I_{N}} + {F_{N}^{T}{\Lambda_{N}\left( {{- 0.5},0} \right)}}} \right){F_{N}.}}}\end{matrix} \right.$ where F_(N) is the forward FFT, F_(n) ^(t) is theinverse FFT and T_(N) is a permutation matrix in the form of:$T_{N} = {\begin{bmatrix}1 & \quad & \quad & \quad \\\quad & \quad & \quad & 1 \\\quad & \quad & \ddots & \quad \\\quad & 1 & \quad & \quad\end{bmatrix}.}$
 36. The computer-implemented process of claim 33wherein the reversible fractional-shifted transform is a 2N-pointreversible fractional-shifted Fast Fourier Transform (FFT) implementedvia a reversible transform S_(2N) of matrix lifting followed by Nreversible rotations K_(2N).
 37. The computer-implemented of claim 36wherein in the fractionally shifted FFT F_(2N)(α,β), with α=β=0.25, thereversible rotations K_(2N) take the form ${K_{2N} = \begin{bmatrix}{\Lambda_{N}\left( {{{\left( {1 + \beta} \right)/2} - \alpha},\alpha} \right)} & \quad \\\quad & {W_{2}^{\beta}{\Lambda_{N}\left( {{{\left( {1 + \beta} \right)/2} - \alpha},\alpha} \right)}}\end{bmatrix}},$ and the reversible transform S_(2N) of matrix liftingis implemented via: ${S_{2N} = {{{{{P_{2N}\begin{bmatrix}I_{N} & \quad \\A_{N} & I_{N}\end{bmatrix}}\begin{bmatrix}B_{N} & \quad \\\quad & I_{N}\end{bmatrix}}\begin{bmatrix}I_{N} & C_{N} \\\quad & I_{N}\end{bmatrix}}\begin{bmatrix}I_{N} & \quad \\D_{N} & I_{N}\end{bmatrix}}Q_{2N}}},$ wherein P_(2N)=I_(2N), Q_(2N)=OE_(2N) and thematrixes in the matrix lifting are: $\quad\left\{ \begin{matrix}{{A_{N} = {{{- \sqrt{2}}{\Lambda_{N}\left( {{- 0.25},0.25} \right)}F_{N}^{T}{\Lambda_{N}\left( {{- 0.25},0} \right)}} - I_{N}}},} \\{{B_{N} = {{- {R_{N}(0.25)}} = \begin{bmatrix}{- 1} & \quad & \quad & \quad \\\quad & \quad & \quad & j \\\quad & \quad & ⋰ & \quad \\\quad & j & \quad & \quad\end{bmatrix}}},} \\{{C_{N} = {{- \frac{1}{\sqrt{2}}}{\Lambda_{N}\left( {{- 0.25},0.25} \right)}F_{N}^{T}{\Lambda_{N}\left( {0.25,0.5} \right)}}},} \\{D_{N} = {{\Lambda_{N}\left( {{- 0.25},0.25} \right)}\left( {\sqrt{2} + {F_{N}^{T}{\Lambda_{N}\left( {{- 0.5},0.125} \right)}}} \right)F_{N}{{\Lambda_{N}\left( {0.25,0} \right)}.}}}\end{matrix} \right.$
 38. The computer-implemented process of claim 21wherein the input signal is encoded in a lossless manner.
 39. Thecomputer-implemented process of claim 21 wherein the input signal isencoded in a progressive-to-lossless manner.